# IGCSE Mathematics Higher: Straight Line Graphs

## Scheme of work: IGCSE Higher: Year 10: Term 4: Straight Line Graphs

#### Prerequisite Knowledge

1. Plotting Straight Line Graphs from a Table of Results
• Concept: Understanding how to create a table of values for a given linear equation and plot the corresponding points on a graph.
• Example: $\text{For } y = 2x + 3, \text{ create a table of values:}$ $\begin{array}{c|c} x & y \\ \hline 0 & 3 \\ 1 & 5 \\ 2 & 7 \\ \end{array}$ $\text{Plot the points } (0,3), (1,5), (2,7) \text{ on a graph.}$
2. Plotting a Straight Line from Two Coordinates
• Concept: Understanding how to plot a straight line graph using two given coordinates.
• Example: $\text{Given points } (1, 2) \text{ and } (3, 6), \text{ plot the line passing through these points.}$
3. Using Straight Line Graphs to Solve Simple Linear Equations
• Concept: Understanding how to use a graph to find the solution to a linear equation.
• Example: $\text{Solve } 2x + 3 = 7 \text{ using the graph of } y = 2x + 3.$ $\text{Find the point where the line intersects } y = 7.$ $\text{The solution is } x = 2.$

## Success Criteria

1. Calculate the Gradient of a Straight Line Given the Coordinates of Two Points
• Objective: Determine the gradient (slope) of a line using the coordinates of two points on the line.
• Example: $\text{Given points } (x_1, y_1) = (2, 3) \text{ and } (x_2, y_2) = (5, 11), \text{ calculate the gradient:}$ $m = \frac{y_2 – y_1}{x_2 – x_1} = \frac{11 – 3}{5 – 2} = \frac{8}{3}$
2. Find the Equation of a Straight Line Parallel to a Given Line
• Objective: Write the equation of a line that is parallel to a given line and passes through a specified point.
• Example: $\text{Given the line } y = 2x + 3 \text{ and a point } (4, 7), \text{ find the equation of the parallel line.}$ $\text{The gradient of the parallel line is the same: } m = 2.$ $\text{Using the point-slope form: } y – y_1 = m(x – x_1)$ $y – 7 = 2(x – 4)$ $y – 7 = 2x – 8$ $y = 2x – 1$
3. Find the Equation of a Straight Line Perpendicular to a Given Line
• Objective: Write the equation of a line that is perpendicular to a given line and passes through a specified point.
• Example: $\text{Given the line } y = \frac{1}{2}x + 3 \text{ and a point } (2, -1), \text{ find the equation of the perpendicular line.}$ $\text{The gradient of the given line is } \frac{1}{2}.$ $\text{The gradient of the perpendicular line is the negative reciprocal: } m = -2.$ $\text{Using the point-slope form: } y – y_1 = m(x – x_1)$ $y + 1 = -2(x – 2)$ $y + 1 = -2x + 4$ $y = -2x + 3$
4. Solve Geometrical Problems Involving Straight Line Graphs
• Objective: Apply the concepts of straight line graphs to solve geometrical problems.
• Example: $\text{For a kite ABCD with coordinates given for points A, B, and C, find the coordinates of point D.}$ $\text{A = (-2, 10), B = (-5.4, 4), C = (4, -5)}$ $\text{Using the properties of a kite and the equations of lines, calculate D.}$

#### Key Concepts

1. Gradient of a Straight Line
• Concept: The gradient (slope) of a straight line measures how steep the line is and is calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate between two points on the line.
• Example: $m = \frac{y_2 – y_1}{x_2 – x_1}$
2. Equation of a Straight Line
• Concept: The equation of a straight line in slope-intercept form is given by $$y = mx + c$$, where $$m$$ is the gradient and $$c$$ is the y-intercept.
• Example: $y = 2x + 3$
3. Parallel Lines
• Concept: Parallel lines have the same gradient but different y-intercepts.
• Example: $\text{If the equation of one line is } y = 2x + 3, \text{ a parallel line has the equation } y = 2x + c \text{ where } c \neq 3.$
4. Perpendicular Lines
• Concept: Perpendicular lines have gradients that are negative reciprocals of each other.
• Example: $\text{If the gradient of one line is } \frac{1}{2}, \text{ the gradient of a perpendicular line is } -2.$
5. Point-Slope Form
• Concept: The equation of a straight line can also be written in point-slope form: $$y – y_1 = m(x – x_1)$$.
• Example: $\text{For a line passing through the point } (2, -1) \text{ with gradient } -2, \text{ the equation is } y + 1 = -2(x – 2).$
6. Geometrical Problems Involving Straight Line Graphs
• Concept: Using properties of shapes and equations of lines to solve geometrical problems.
• Example: $\text{For a kite with vertices } A, B, C, \text{ and } D, \text{ use the properties of the kite and coordinates to find } D.$

## Common Mistakes

1. Calculating the Gradient of a Straight Line
• Common Mistake: Students might reverse the coordinates when calculating the gradient, leading to incorrect results.
• Example: $\text{Incorrect: } m = \frac{x_2 – x_1}{y_2 – y_1}$ $\text{Correct: } m = \frac{y_2 – y_1}{x_2 – x_1}$
2. Finding the Equation of a Parallel Line
• Common Mistake: Students might use the same y-intercept as the given line instead of finding the correct one for the new line.
• Example: $\text{Incorrect: If the given line is } y = 2x + 3, \text{ the parallel line is } y = 2x + 3.$ $\text{Correct: If the given line is } y = 2x + 3 \text{ and the new line passes through } (4, 7), \text{ then } y = 2x – 1.$
3. Finding the Equation of a Perpendicular Line
• Common Mistake: Students might forget to take the negative reciprocal of the gradient when finding the equation of a perpendicular line.
• Example: $\text{Incorrect: If the gradient of the given line is } \frac{1}{2}, \text{ the gradient of the perpendicular line is } \frac{1}{2}.$ $\text{Correct: If the gradient of the given line is } \frac{1}{2}, \text{ the gradient of the perpendicular line is } -2.$
4. Using the Point-Slope Form
• Common Mistake: Students might incorrectly substitute the coordinates into the point-slope form equation.
• Example: $\text{Incorrect: For a point } (2, -1) \text{ and gradient } -2, \text{ the equation is } y – 1 = -2(x + 2).$ $\text{Correct: For a point } (2, -1) \text{ and gradient } -2, \text{ the equation is } y + 1 = -2(x – 2).$
5. Solving Geometrical Problems Involving Straight Line Graphs
• Common Mistake: Students might incorrectly apply the properties of shapes or the equations of lines when solving geometrical problems.
• Example: $\text{Incorrect: Assuming a kite’s diagonals are always equal in length.}$ $\text{Correct: Using the properties of a kite (two pairs of adjacent sides are equal) and correct line equations to find missing coordinates.}$

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